Question

Describe the Ito process and discuss, giving details, why it is more applicable to describing the stochastic process of asset prices, compared to both the basic Weiner process and the generalised Weiner process.

Special Requirement: Min 400 words

Answer 1

stochastic (Ito) integrals are used in finance to model the gains realized by trading.

There are two principle ideas at play here that enable Ito integrals to be used for this purpose:

1. The concept of a buy-and-hold strategy can be formulated as an integrand.
2. All trading strategies are suitably approximated by buy-and-hold strategies.

In finance terms, a buy-and-hold strategy means that we buy a stock at time uu, and then hold it until time ww, at which time we sell it. What would be the trading gains from using this strategy? To begin, let’s define XtXt as the random process which models the evolution of stock prices through time.

Ok, if we use this buy and hold strategy, how much profit do we make? We buy the stock at price XuXu, and sell it at XwXw, so our profit is equal to Xw−XuXw−Xu. Note that this quantity is a random variable: our profits are indeed random, because the stock is.

We can formally define the buy and hold strategy as H0t:=1([u,w])(t)Ht0:=1([u,w])(t). Note that this is a (deterministic) process, equal to 1 between u and w, and 0 elsewhere. At this point, we formally define the integral ∫T0H0tdXt:=Xw−Xu∫0THt0dXt:=Xw−Xu. We are basically saying, you get a difference when you integrate a buy-and-hold strategy, also known as a simple integrand. You might object to this formal definition, but that’s also how Riemann integration is defined (simple integration is tautological or “obvious”, and it is built up to more complicated integrands).

At this point, you can see how we might extend our buy-and-hold to other, more complicated strategies. For example, we might buy three units of stock, or we might buy and hold from time u to w, and then buy and hold again from time x to z. Any type of sum of step functions can be “integrated”. We can also extend past deterministic step functions, to use random times which don’t cheat by looking into the future. These are the so called stopping times. For example, buy the stock the first time it hits a price of ten, and sell it when it hits a price of fifteen (of course these times can be infinite!)

In practice, these types of composed buy-and-hold strategies are the ones which can be implemented on a computer. However, mathematical models require the use of trading strategies which vary continuously. Indeed, in the Black-Scholes model, we care a lot about hedging strategies, which enable us to eliminate all the risk in a European option’s payout through the use of clever trading in the underlying security; these hedging strategies, as theoretical objects, tend to vary continuously. Furthermore, it’s nice to be able to have a good theory for such objects, because they ensure good behavior of the buy-and-hold strategies, as they get more and more complex.

The nice thing about Ito integration, is that you can suitably approximate continuously varying strategies by buy-and-hold strategies (re: L^2 norm, Ito Isometry). This allows us to bootstrap our intuition about buy-and-hold strategies to the full generality of any continuous strategy. Put another way, given a continuously varying strategy, we can discretize it, to get an arbitrarily good approximation, and this discretized strategy can be represented as a buy-and-hold strategy, albeit one that enters and exits at non-deterministic times, and purchases non-deterministic amounts of stocks (depending on how things play out).

Finally, the nature of our choice of integrator (XtXt) forces us to use Ito Integration instead of a simpler L^1 integration theory. This is fundamentally a consequence of Brownian Motion having very rough paths of infinite variation. For an integration theory to be coherent, its fundamental property is the type of approximation by buy-and-hold integrands that we discussed above, and if we try Riemann or Lebesgue integration with Brownian paths, we will usually fail.

The Wiener process is at least (1) weakly stationary, and I believe it does not meet the requirements for (2) strict stationarity.

(1) It is weakly stationary because its covariance is the same for any two equal-length intervals. This is implied by its definition, that is has stationary increments.

(2) It is not strictly stationary, however, because as t increases, so does its variance. This is evident (a) graphically, and (b) mathematically.

(2.a.) A plot of a Wiener process shows that as t increases its values can be more negative or more positive. Obviously it is more varied.

(2.b.)

$$

var(B(t_i)) = t_i < var(B(t_{i+1}) = t_{i+1}

$$

For $0<t_i<t_{i+1}<\infty$